lons of different momentum (p), having the same charge, enter normally a uniform magnetic field. The radius of the orbit of an ion is proportional to

Options

A. `P`
B. `1/p`
C. `P^2`
D. `1/p^2`

Answer

To solve the problem, we need to find the relationship between the radius of the orbit of an ion moving in a magnetic field and its momentum, given that the ions have the same charge.

### Step-by-Step Solution:

1. **Understanding the Motion of Charged Particles in a Magnetic Field**:
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts perpendicular to both the velocity of the particle and the magnetic field. This force causes the particle to move in a circular path.

2. **Expression for Magnetic Force**:
The magnetic force \( F \) acting on a charged particle moving with velocity \( v \) in a magnetic field \( B \) is given by:
\[ F = qvB \]
where \( q \) is the charge of the particle.

3. **Centripetal Force**:
For a particle moving in a circular path of radius \( r \), the centripetal force required to keep it in that path is provided by the magnetic force:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle.

4. **Equating the Forces**:
Since the magnetic force provides the centripetal force, we can set the two expressions equal to each other:
\[ qvB = \frac{mv^2}{r} \]

5. **Rearranging the Equation**:
Rearranging the equation to solve for the radius \( r \):
\[ r = \frac{mv}{qB} \]

6. **Substituting Momentum**:
The momentum \( p \) of the particle is defined as:
\[ p = mv \]
Therefore, we can substitute \( mv \) in the radius equation:
\[ r = \frac{p}{qB} \]

7. **Proportionality**:
Since the charge \( q \) and the magnetic field \( B \) are constants (as given in the problem that the ions have the same charge), we can conclude that the radius \( r \) is directly proportional to the momentum \( p \):
\[ r \propto p \]

### Conclusion:
The radius of the orbit of an ion in a magnetic field is directly proportional to its momentum.